Theorem dfon2lem1 35975

Description: Lemma for dfon2 35984. (Contributed by Scott Fenton, 28-Feb-2011.)

Assertion

Ref	Expression
dfon2lem1	⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}

Proof of Theorem dfon2lem1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		truni 5220	. 2 ⊢ (∀𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}Tr 𝑦 → Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)})
2		nfsbc1v 3760	. . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		nfv 1915	. . . . 5 ⊢ Ⅎ𝑥Tr 𝑦
4		nfsbc1v 3760	. . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜓
5	2, 3, 4	nf3an 1902	. . . 4 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓)
6		vex 3444	. . . 4 ⊢ 𝑦 ∈ V
7		sbceq1a 3751	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8		treq 5212	. . . . 5 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
9		sbceq1a 3751	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ [𝑦 / 𝑥]𝜓))
10	7, 8, 9	3anbi123d 1438	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ Tr 𝑥 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓)))
11	5, 6, 10	elabf 3630	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓))
12	11	simp2bi 1146	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} → Tr 𝑦)
13	1, 12	mprg 3057	1 ⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}

Syntax hints:   ∧ w3a 1086   ∈ wcel 2113  {cab 2714  [wsbc 3740  ∪ cuni 4863  Tr wtr 5205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-v 3442  df-sbc 3741  df-ss 3918  df-uni 4864  df-iun 4948  df-tr 5206

This theorem is referenced by:  dfon2lem3  35977  dfon2lem7  35981